A kinetic model for coagulation–fragmentation

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www.elsevier.com/locate/anihpc

A kinetic model for coagulation–fragmentation

Damien Broizat

Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, F-06108 Nice cedex 02, France Received 16 April 2009; received in revised form 25 November 2009; accepted 25 November 2009

Available online 3 December 2009

Abstract

The aim of this paper is to show an existence theorem for a kinetic model of coagulation–fragmentation with initial data satisfying the natural physical bounds, and assumptions of finite number of particles and finiteL^p-norm. We use the notion of renormalized solutions introduced by DiPerna and Lions (1989) [3], because of the lack ofa prioriestimates. The proof is based on weak- compactness methods inL¹, allowed byL^p-norms propagation.

1. Introduction

Coalescence and fragmentation are general phenomena which appear in dynamics of particles, in various fields (polymers chemistry, raindrops formation, aerosols, . . . ). We can describe them at different scales, which lead to different mathematical points of view. First, we can study the dynamics at the microscopic level, with a system ofN particles which undergo successives mergers/break ups in a random way. We refer to the survey [1] for this stochastic approach. Another way to describe coalescence and fragmentation is to consider the statistical properties of the system, introducing the statistical distribution of particlesf (t, m)of massm >0 at timet0 and studying its evolution in time. This approach is rather macroscopic. But we can put in an intermediate level, by considering a densityf which depends on more variables, like positionxor velocityvof particles, and this description is more precise. Here, we start by discussing models with density, from the original (withf =f (t, m)) to the kinetic one (withf=f (t, x, m, v)), which is the setting of this work.

Depending on the physical context, the mass variable is discrete (polymers formation) or continuous (raindrops formation). It leads to two sorts of mathematical models, withm∈Norm∈(0,+∞), but we focus on the continuous case. To understand the relationship between discrete and continuous equations, see [16].

1.1. The original model

The discrete equations of coagulation have been originally derived by Smoluchowski in [21,22], by studying the Brownian motion of colloidal particles. It had been extended to the continuous setting by Müller [20], giving the

E-mail address:broizat@unice.fr.

doi:10.1016/j.anihpc.2009.11.014

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following mathematical model, called theSmoluchowski’s equation of coagulation:

∂f

∂t(t, m)=Q⁺_c(f, f )−Q⁻_c(f, f ), (t, m)∈(0,+∞)². (1.1) This equation describes the evolution of the statistical mass distribution in time. At each timet >0, the termQ⁺_c(f, f ) represents the gain of particles of massmcreated by coalescence between smaller ones, by the reaction

m +

m−m

→ {m}.

The term Q⁻_c(f, f )is the depletion of particles of massmbecause of coagulation with other ones, following the reaction

{m} + m

→

m+m . Namely, we have

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

Q⁺_c(f, f )(t, m)=1 2

m 0

A m, m−m

f t, m

f t, m−m dm,

Q⁻_c(f, f )(t, m)= +∞

0

A m, m

f (t, m)f t, m dm,

whereA(m, m)is the coefficient of coagulation between two particles, which governs the frequency of coagulations, according to the mass of clusters. In his original model, Smoluchowski derived the following expression forA:

A m, m

= m^1/3+m^1/3 m⁻^1/3+m⁻^1/3

. (1.2)

In many cases, coalescence is not the only mechanism governing the dynamics of particles, and other effects should be taken into account. A classical phenomenon which also occurs is the fragmentation of particles in two (or more) clusters, resulting from an internal dynamic (we do not deal here with fragmentation processes induced by particles collisions). This binary fragmentation is modeled by linear additional reaction terms in Eq. (1.1), namely

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

Q⁺_f(f )(t, m)= +∞

m

B m, m

f t, m dm,

Q⁻_f(f )(t, m)=1

2f (t, m)B₁(m), whereB₁ m

=

m

0

B m, m dm.

The functionB(m, m) is the fragmentation kernel, it measures the frequency of the break-up of a massm in two clustersmandm−m, form < m. So, at each timet, the termQ⁺_f(f )is the gain of particles of massm, resulting from the following reaction of fragmentation:

m

→ {m} +

m−m ,

whereas Q⁻_f(f )stands for the loss of particles of massm, because of a break-up into two smaller pieces, by the following way:

{m} → m

+

m−m

, withm< m.

Thus, the continuouscoagulation–fragmentation equationwrites

∂f

∂t(t, m)=Q⁺_c(f, f )−Q⁻_c(f, f )+Q⁺_f(f )−Q⁻_f(f ), (t, m)∈(0,+∞)². (1.3) In the 90’s, many existence and uniqueness results have been proved about this problem, see for instance [23], [24], or [13] for an approach by the semigroups of operators theory. These results are true under various growth hypotheses

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on kernelsAandB, but these assumptions often allow unbounded kernels, which is important from a physical point of view.

However, this coagulation–fragmentation model do not take the spatial distribution of particles into account. This leads to “spatially inhomogeneous” mathematical models, where the density of particlesf (t, x, m)depends also of a space variablex∈R³.

1.2. Spatially inhomogeneous models

A first example consists of diffusive models, corresponding to the situation where particles follow a Brownian motion at the microscopic scale, with a positive and mass-dependent coefficient of diffusiond(m). From a physical point of view, it implies that particles are sufficiently small to undergo the interaction with the medium, i.e. the shocks with the molecules of the fluid in which the particles evolve. In the statistical description, a spatial-Laplacian appears, giving thediffusive coagulation–fragmentation equation:

∂f

∂t(t, x, m)−d(m)_xf (t, x, m)=Q⁺_c(f, f )−Q⁻_c(f, f )+Q⁺_f(f )−Q⁻_f(f ),

(t, x, m)∈(0,+∞)×R³×(0,+∞). (1.4)

We refer to [15] for a global existence theorem for the discrete diffusive coagulation–fragmentation equation inL¹, and to [17] for the continuous one, improved in [19] (with less restrictive conditions on the kernels), then in [2] (with uniqueness of the solution).

The second way to correct the spatially homogeneous problem is to assume that the particles are transported with a deterministic velocityv. At the statistical level, this adds a linear transport termv.∇xf to Eq. (1.3). This velocity can be a given velocityv=v(t, x, m)or the inner velocity of the particles. The first case has been studied in [6], with an existence and uniqueness theorem, and furthermore the continuous dependence on the initial data. Physically, it corresponds to the dynamics of particles with rather low mass which follow a velocity drift depending only on the surrounding fluid. In the second case, particles are also identified by their momentump∈R³ in addition to their massm(withv=p/m): we have a kinetic model, which is relevant to describe the dynamics of particles of varying size/mass according to coagulation/fragmentation events, like in aerosols. At the microscopic scale, the coagulation/fragmentation processes become “multi-dimensional”, with mass-momentum conservation at each merger/break up according to the following scheme:

Coagulation: {m} + m

→ m

, {p} +

p

→ p

,

Fragmentation:

m

→ {m} + m ,

p

→ {p} + p

, wherem:=m+m,m >0,m>0, andp:=p+p.

Thus, in the statistical description, the density depends on time, position, mass and momentum:f =f (t, x, m, p).

But even if this kind of kinetic models provides a rather good description of phenomena, it is harder to study, so there are less results than for the diffusive ones. Moreover, it is difficult to know the exact physical form of the kernels. And finally, the numerical aspects are a real problem on these models: because of a high dimension (at least 7 plus time), it seems to be very difficult, maybe impossible, to compute the solutions on a long time.

Concerning the results, a global existence theorem for the sole coagulation has been demonstrated in [7]. The proof is based onL^p-norms dissipation for any formal solution, and on weak-compactness methods inL¹. This result has been extended to a more general class of coalescence operators in [12] (but under stronger restriction on the initial data), with a very different method of proof. For the sole fragmentation, a difficulty is due to the blow-up of kinetic energy, which grows at each microscopic event. Thus, it is reasonable to take the internal energy of particles into account, which balances the gain of kinetic energy during a break up. With that modeling, the work [11] provides global existence for a kinetic fragmentation model, with general growth assumptions on the kernelB, by using correct entropies.

The aim of this work is to combine both of these analysis. We deal with assumptions which are similar to [7], but a little bit more restrictive concerning the kernelA. The obtaining ofa prioriestimates is strongly inspired from [7],

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with a big difference however. The authors obtained refined estimates, including a dissipative quadratic term thanks to which coagulation bilinear terms make sense, but which is unfortunately not present here because of the balance problems between coagulation and fragmentation operators. Thus, this lack of estimates does not allows us to define well the reaction term of coagulation with only the a priori bounds (specifically, we cannot say that the bilinear loss termQ⁻_c(f, f )lies inL¹_loc, as it is shown in Subsection 3.2). That is why we use the DiPerna–Lions theory of renormalized solutions, introduced in [3] to show global existence for Boltzmann equation, which presents similar problems.

1.3. Description of the kinetic model and outline of the paper

Now, let us describe precisely the model we study. The parameters which describe the state of a particle are denoted by

y:=(m, p, e)∈Y:=(0,+∞)×R³×(0,+∞),

mfor the mass,pthe impulsion, andethe internal energy. At the microscopic scale, coalescence and fragmentation conserve total energy (kinetic energy +internal energy), thus we can compute the internal energy of daughter(s) particle(s).

Coagulation: {e} + e

→ e

. We have

|p|²

2m +e+|p|²

2m +e= |p+p|² 2(m+m)+e, thus

e=e+e+E₋ m, m, p, p

, whereE₋ m, m, p, p

:= |mp−mp|² 2mm(m+m)0 (E₋is the loss of kinetic energy resulting from the merger).

Fragmentation:

e

→ {e} + e

. We have

|p|²

2m +e=|p|²

2m +e+ |p−p|² 2(m−m)+e, thus

e=e−e−E₊ m, m, p, p

, whereE₊ m, m, p, p

:= |mp−mp|² 2mm(m−m)0 (E₊is the gain of kinetic energy resulting from the break up).

Remark 1.1.Let us point out the following symmetries:

E₋ m, m, p, p

=E₋ m, m, p, p

and E₊ m, m, p, p

=E₊ m, m−m, p, p−p ,

and the relation: E₋(m, m, p, p)=E₊(m+m, m, p+p, p), which is consistent with the two phenomena’s reciprocity.

We use the following notations:

• ify=(m, p, e),y=(m, p, e), then we denote

y:=y+y:= m+m, p+p, e+e+E₋ m, m, p, p ,

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• ify=(m, p, e),y=(m, p, e), withm < m ande < e−E₊(m, m, p, p), then we say thaty < y and we denote

y:=y−y:= m−m, p−p, e−e−E₊ m, m, p, p .

With this formalism, we naturally have(y−y)+y=y, but note carefully thaty < yis not an order relation onY. Remark 1.2. For ally∈Y,{y∈Y, y < y} ⊂(0, m)×B√

2me+|p|² ×(0, e). DenotingY_R :=(0, R)×B_R× (0, R)⊂Y, we have

y < y, y∈Y_R ⇒ y∈(0, R)×B^√_3R×(0, R)⊂Y_2R. (1.5) Finally, we point out that the map (m, m, p, p, e, e)→(m, m, p, p, e, e)is a diffeomorphism with C^∞ regularity within the domain

0< m < m, p, p∈R³, 0< e < e−E₊ m, m, p, p

⊂Y² which preserves volume.

We denote by f (t, x, m, p, e)=f (t, x, y) the particles density, which is a nonnegative function depending on timet0, positionx∈R³, and the mass-momentum-energy variabley. To shorten the notations, we set for eacht,x, f (y)=f (t, x, y), orf=f (t, x, y),f=f (t, x, y), andf=f (t, x, y). The complete model then reads:

∂_tf +p

m.∇xf =Q⁺_c(f, f )−Q⁻_c(f, f )+Q⁺_f(f )−Q⁻_f(f ), t∈(0,+∞), x∈R³, y=(m, p, e)∈Y,

(ECF) with ⎧

⎪⎪

⎨

⎪⎪

⎩

Q⁺_c(f, f )(y)=1 2

Y

A y, y−y f y

f y−y

1_{y<y}dy,

Q⁻_c(f, f )(y)=f (y)Lf (y), Lf (y):=

Y

Ay, y f y

dy,

and ⎧

⎪⎪

⎨

⎪⎪

⎩

Q⁺_f(f )(y)=

Y

B y, y f y

1_{_y>y}dy,

Q⁻_f(f )(y)=1

2B1(y)f (y), B1 y :=

Y

B y, y

1_{_y<y}dy.

FunctionsAetB are respectively the coagulation and fragmentation kernels. They are nonnegative functions, inde- pendent of(t, x), which satisfy the natural properties of symmetry:

∀ y, y

∈Y², Ay, y

=A y, y

, (1.6)

∀ y, y

∈Y², y < y, B y, y

=B y, y

. (1.7)

The kernelA(y, y)represents the coalescence rate between two particlesyandy, whereasB(y, y)is the fragmentation rate for a particleywhich breaks in two clustersyandy.

We assume thatAfulfills the following structure assumption:

∀ y, y

∈Y², Ay, y

A y, y

+A y, y

. (1.8)

Remark 1.3.We can insist on the fact that this assumption is more general than the classicalGalkin–Tupchiev mono- tonicity condition:

∀y < y, A y, y−y

A y, y

. (1.9)

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In the “mono-dimensional” case, the Smoluchowski kernel given by (1.2)do not satisfy (1.9) but satisfies(1.8), that’s why the first existence result established in [17] under Galkin–Tupchiev condition was extended in [19] to kernels which satisfy(1.8)only.

We also require thatAandBhave a mild growth:

∀R >0,

YR

A(y, y)

|y| dy −→

|y|→+∞0, (1.10)

∀R >0,

Y_R

B(y, y)

|y| 1_{y<y}dy −→

|y|→+∞0, (1.11)

andBis truncated as:

∃C0>1,

⎧⎪

⎪⎨

⎪⎪

⎩

m> C0m or e+|p|²

2m > C₀

e+|p|² 2m

⇒ B y, y

=0. (1.12)

Remark 1.4.The physical interpretation of this truncature assumption is to prevent the creation of too small clusters compared to the mother particle. From a mathematical point of view, it allows the total number of particles (the L¹-norm off) to be finite at each timet >0.

We also need to haveB1locally bounded:

∀R >0, B₁∈L^∞(Y_R), (1.13)

as well asA:

∀R >0, A∈L^∞ Y_R²

. (1.14)

Remark 1.5. Unfortunately, these assumptions of growth and boundedness are more restrictive, and in the mono- dimensional case, the Smoluchowski kernel (1.2) doesn’t satisfy them any more. The examples given in [7] for the sole coagulation, namely

A m, m, p, p

= m^α+m^α2 p

m−p m

, 0α <1/2,

(for the dynamics of liquid droplets carried by a gaseous phase) or A m, m, p, p

=

m+m mm

α p

m−p m

^γ, 0α1, −3< γ 0

(for a stellar dynamics context) do not fit neither. Here, we need coalescence kernels which are bounded when m, m→0. But it is difficult to know the exact physical form of the kernelsA andB because of the complexity of this kinetic model. Nevertheless, simple kernels given byA(m, m)=m^α+m^αwith 0< α <1 fit.

Finally, we assume thatAcontrolsBin the following sense:

∃s >1, ∃0< δ < 1 6s−5<1,

∀y∈Y,

Y

B(y, y)^s

A(y, y)^s⁻¹1_{y<y}dy1+m+|p|²

2m +e+1 2B1 yδ

. (1.15)

Remark 1.6.This last assumption is more technical, but seems necessary to balance the contributions of the interaction termsQ_c(f, f )andQ_f(f ), which are difficult to compare becauseQ_c(f, f )is quadratic whereasQ_f(f )is linear.

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The paper consists in the proof of the following theorem.

Theorem 1.1.LetAandB be kernels satisfying(1.6)–(1.8)and(1.10)–(1.15)and letf⁰be a nonnegative initial data which satisfies

K f⁰ :=

R³×Y

1+m+|p|²

2m +e+m|x|²

f⁰(x, y)+f⁰(x, y)^s

dx dy <∞, (1.16)

then for allT >0, there existsf ∈C([0, T], L¹(R³×Y ))such thatf (0)=f⁰andf is a renormalized solution to (ECF). Moreover,

a.e. t∈(0, T ),

R³×Y

1+m+|p|²

2m +e+m|x|²

f (t, x, y) dx dyK_T, (1.17)

a.e. t∈(0, T ),

R³×Y

f (t, x, y)^sdx dyK_T, (1.18)

where the constantK_T depends only onC₀,T,K(f⁰),sandδ(defined in(1.12)and(1.15)).

Beyond existence problems, there are lots of others interesting subjects to explore. A first one concerns the mass conservation of the solutionf, which is still an open problem for such kinetic models, even for the case of the sole coagulation. In the spatially homogeneous case, it has been shown in [5] that total mass is preserved in time under mild growth hypotheses on kernels. But we know that in case of strong coagulation (typically the case of multiplicative kernels), a phenomenon of relation occurs, which force the total mass of the system to decay from a certain time T_g<+∞. Then, problems of convergence to an equilibrium have been already studied for the spatially homogeneous equation [18], under a detailed balance condition between kernelsAandB. We can also mention existence of self- similar solutions [8,9,14], always for the spatially homogeneous case.

In a first section, we will derive thea prioriestimates from the equation, giving the proper setting of the problem.

Then, the proof of theorem is based on a well-known stability principle which says that if we are able to pass to the limit in the equation (the set of solutions is closed in a certain sense), then it would be easy to show the existence of a solution, applying the stability result to a sequence of approached problems which we can solve. So, the aim of the last section is to prove rigorously such a stability result and in fact, we work in the context of renormalized solutions, because the reaction term cannot be defined as a distribution simply using thea prioriestimates.

1.4. Different notions of solutions

We discuss here on different notions of solutions, recalling the DiPerna–Lions results. We set Q(f, f )= Q⁺_c(f, f )−Q⁻_c(f, f )+Q⁺_f(f )−Q⁻_f(f ).

Definition 1.2.Letf be a nonnegative function, such thatf ∈L¹_loc((0,+∞)×R³×Y ). We say thatf is a renormalized solution of (ECF) if

Q^±_c(f, f )

1+f ∈L¹_loc (0,+∞)×R³×Y ,

Q^±_f(f )

1+f ∈L¹_loc (0,+∞)×R³×Y , and if the functiong:=log(1+f )satisfies the renormalized equation

∂_tg+p

m.∇xg=Q(f, f )

1+f (ECFR)

inD((0,+∞)×R³×Y ).

The renormalization makes passing to the limit impossible because of the quotients in the reaction term, that is why we also need another notion of solution: the mild solutions, which only require local integrability in time and provide Duhamel’s integral formulations to the problem in which we are able to pass to the limit.

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Definition 1.3. Letf be a nonnegative function, such thatf ∈L¹_loc((0,+∞)×R³×Y ). We say thatf is a mild solution of (ECF) if for almost all(x, y)∈R³×Y,

∀T >0, Q^±_c(f, f )(t, x, y)∈L¹ (0, T )

, Q^±_f(f )(t, x, y)∈L¹ (0, T ) , and

∀0< s < t <∞, f(t, x, y)−f(s, x, y)= t s

Q(f, f )(σ, x, y) dσ, (1.19)

wherehdenotes the restriction to the characteristic lines of the equation:

h(t, x, m, p, e):=h

t, x+tp

m, m, p, e

.

The following results are proved in [3]:

Lemma 1.4.

(i) IfQ^±_c(f, f )∈L¹_loc((0,+∞)×R³×Y )andQ^±_f(f )∈L¹_loc((0,+∞)×R³×Y ),then the following assertions are equivalent:

• f is a solution of (ECF)in the sense of distributions,

• f is a renormalized solution of (ECF),

• f is a mild solution of (ECF).

(ii) Iff is a renormalized solution of (ECF), then for all functionβ∈C¹([0,+∞))such that|β(u)|₁^C₊_u, the composed functionβ(f )is a solution of

∂_tβ(f )+p

m.∇xβ(f )=β(f )Q(f, f )

in the sense of distributions(here, the right side lies inL¹_loc((0,+∞)×R³×Y )).

(iii) f is a renormalized solution of (ECF)if and only iff is a mild solution of(ECF), Q^±_c(f, f )

1+f ∈L¹_loc (0,+∞)×R³×Y and

Q^±_f(f )

1+f ∈L¹_loc (0,+∞)×R³×Y .

2. A prioriestimates

We consider the Cauchy problem (ECF),

f (0, x, y)=f⁰(x, y). (2.20)

We suppose in this section that (2.20) admit a sufficiently smooth solutionf in order to handle some formal quantities which are conserved or propagated by Eq. (ECF). More precisely, we will show the propagation ofL^qbounds for the solution along time:

Proposition 2.1.If the initial dataf⁰satisfies K f⁰

:=

R³×Y

1+m+|p|²

2m +e+m|x|²

f⁰(x, y)+f⁰(x, y)^s

dx dy <∞, (2.21)

then for allT >0, any classical solution of the Cauchy problem(2.20)satisfies sup

t∈[0,T] R³×Y

1+m+|p|²

2m +e+m|x|²

f (t, x, y)+f (t, x, y)^q

dx dyK_T, (2.22)

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for all the exponentsq∈(5/6, s], and also T

0

R³

D1 f (t, x)

+D2 f (t, x)

dx dtKT, (2.23)

where

D₁ f (t, x) :=1

2

Y×Y

A y, y

sup f, f

inf f, fs

dydy0, (2.24)

D₂ f (t, x)

:=s−δ 2

Y×Y

B y, y

f t, x, ys

1_{y<y}dydy0, (2.25)

and the constantK_T depends only onC₀,T,K(f⁰),sandδ.

2.1. Basic physical estimates

We start with a fundamental formula, which gives the variation in time of some integral quantities involving the solutionf.

Lemma 2.2.LetH (u)be a function withC¹regularity on[0,+∞)andΦ(y)a real or vectorial function. We have d

dt

R³×Y

Φ(y)H f (t, x, y) dx dy

=1 2

R³×Y×Y

Aff Φd_uH f

−Φ d_uH (f )−Φd_uH f

dydy dx

+1 2

R³×Y×Y

Bf Φ d_uH (f )+Φd_uH f

−Φd_uH f

1_{y<y}dy dydx, (2.26)

whereduH=^dH_du.

Proof. Using (ECF), we have d

dt

R³×Y

Φ(y)H (f ) dx dy=

R³×Y

Φ(y) duH (f ) ∂tf dy dx

=

R³×Y

Φ(y) d_uH (f ) Q⁺_c(f, f )−Q⁻_c(f, f ) dy dx

+

R³×Y

Φ(y) d_uH (f ) Q⁺_f(f )−Q⁻_f(f ) dy dx

−

R³×Y

divx

−Φ(y)H (f )p m

dy dx.

The integral with divergence vanishes thanks to Stokes’ formula. Whence d

dt

R³×Y

Φ(y)H (f ) dx dy=

R³×Y

Φ(y) duH (f ) Q⁺_c(f, f )−Q⁻_c(f, f ) dy dx

+

R³×Y

Φ(y) d_uH (f ) Q⁺_f(f )−Q⁻_f(f ) dy dx.

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Using Fubini’s theorem (formally), we can write d

dt

R³×Y

Φ(y)H (f ) dx dy=1 2

R³×Y×Y

Φ(y) d_uH (f ) A y, y−y f y

f y−y

1_{y<y}dydy dx

−

R³×Y×Y

Φ(y) d_uH (f ) A y, y

f (y)f y

dydy dx

+

R³×Y×Y

Φ(y) d_uH (f ) B y, y f y

1_{y>y}dydy dx

−1 2

R³×Y

Φ y

d_uH f B₁ y

f y dydx.

If we change variables(y, y−y)→(y, y)in the first integral, we obtain d

dt

R³×Y

Φ(y)H (f ) dx dy=1 2

R³×Y×Y

Φ y+y

duH f y+y

A y, y f y

f (y) dydy dx

−

R³×Y×Y

Φ(y) duH (f ) A y, y

f (y)f y

dydy dx

+

R³×Y×Y

1_{_y>y}dydy dx

−1 2

R³×Y×Y

Φ y

duH f

B y, y f y

1_{y<y}dy dydx.

The symmetry ofAallows us to write

R³×Y×Y

Φ(y) d_uH (f ) A y, y

f (y)f y

dydy dx

=1 2

R³×Y×Y

Φ(y) d_uH (f ) A y, y

f (y)f y

dydy dx

+1 2

R³×Y×Y

Φ y

duH f

A y, y

f (y)f y

dydy dx,

using the change of variables(y, y)→(y, y).

The same applies toB with(y, y)→(y, y−y):

R³×Y×Y

1_{y>y}dydy dx

=1 2

R³×Y×Y

1_{y>y}dydy dx

+1 2

R³×Y×Y

Φ y−y

d_uH f y−y

B y, y f y

1_{y>y}dydy dx. 2

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Applying this lemma withH (u)=u, it gives d

dt

R³×Y

Φ(y)f dx dy=1 2

R³×Y×Y

Aff Φ−Φ−Φ

dydy dx

+1 2

R³×Y×Y

Bf Φ+Φ−Φ

1_{y<y}dydy dx. (2.27)

ChoosingΦ(y)=m, we obtain mass conservation:

d dt

R³×Y

mf (t, x, y) dx dy=0. (2.28)

WithΦ(y)=p, we also get the momentum conservation:

d dt

R³×Y

pf (t, x, y) dx dy=0. (2.29)

Then, choosingΦ(y)=^|_2m^p^|²+e, we recover the total energy conservation:

d dt

R³×Y

|p|² 2m +e

f (t, x, y) dx dy=0. (2.30)

Moreover, we can control space momenta:

Lemma 2.3.For allT >0, there exists a constantC_T >0such that

∀t∈ [0, T],

R³×Y

m|x|²f (t, x, y) dx dyC_T. (2.31)

Proof. In view of Eq. (ECF) and the Stokes formula, we have d

dt

R³×Y

m|x|²f dx dy= −

R³×Y

|x|²p.∇xf dx dy

=2

R³×Y

x.pf (t, x, y) dx dy

2

R³×Y

m|x|²f dx dy

1/2

R³×Y

|p|² m f dx dy

1/2

,

and we conclude with (2.30) and Gronwall’s lemma. 2 Finally, we can control the number of particles in finite time:

Lemma 2.4.We set N₀:=

R³×Y

f⁰(x, y) dx dy, M₀:=

R³×Y

mf⁰(x, y) dx dy,

E₀:=

R³×Y

|p|² 2m +e

f⁰(x, y) dx dy.

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Then, there exists a constantC >0depending only onC₀such that

∀T >0, ∀t∈ [0, T],

R³×Y

f (t, x, y) dx dy N0+CT (M0+E0)

e^CT+M0+E0. (2.32)

Proof. We use formula (2.27) withΦ(y)=1

{m1, e+^|^p_2m^|²1}. SinceΦis nonnegative and subadditive in the sense of coalescence (i.e.ΦΦ+Φ), we have

d dt

R³×Y

1{m1, e+^|2m^p^|²1}f dy dx1 2

R³×Y×Y

Bf Φ+Φ−Φ

1_{y<y}dy dydx

=

R³×Y×Y

Bf

Φ−Φ 2

1_{y<y}dy dydx

R³×Y×Y

BfΦ1_{_y<y}dy dydx

=

R³×Y×Y

Bf1

{y<y, m1, e+^|p|_2m²1}dy dydx.

In the last integral, if m> C₀, then, sincem1, we haveB(y, y)=0 according to assumption (1.12). The same applies ife+^|_2m^p^|² > C₀. Thus,

d dt

R³×Y

f (t, x, y)1

{m1, e+^|^p_2m^|²1}dy dx

R³×Y Y

B y, y

1_{y<y}dy

f t, x, y

1{mC0, e+^|^p|_2m²C0}dydx

=

R³×Y

B₁ y

f t, x, y

1{mC₀, e+^|p|_2m²C₀}dydx.

DenotingC:=sup_y∈Y_2C

0B₁(y), we obtain d

dt

R³×Y

f (t, x, y)1

{m1, e+^|^p_2m^|²1}dy dx

C

R³×Y

f (t, x, y) dy dx

C

R³×Y

f (t, x, y)1

{m1, e+^|2m^p^|²1}dy dx+

R³×Y

mf (t, x, y) dy dx

+

R³×Y

e+|p|²

2m

f (t, x, y) dy dx

.

Using (2.28) and (2.30), we have d

dt

R³×Y

f (t, x, y)1

{m1, e+^|p|2m²1}dy dxC

R³×Y

f (t, x, y)1

{m1, e+^|p|2m²1}dy dx+C(M₀+E₀).

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We integrate this inequality in time. Then, Gronwall’s lemma provides

∀T >0, ∀t∈ [0, T],

R³×Y

f (t, x, y)1

{m1, e+^|p|_2m²1}dy dx N₀+CT (M₀+E₀) e^CT.

We conclude noting that

R³×Y

f (t, x, y) dy dx

R³×Y

f (t, x, y)1

{m1, e+^|_2m^p^|²1}dy dx+

R³×Y

m+e+|p|² 2m

f (t, x, y) dy dx,

and using (2.28) and (2.30) again. 2

To summarize, if we setE(x, y)=1+m+^|_2m^p^|² +e+m|x|², and if we suppose that the initial data satisfies K f⁰

:=

R³×Y

E(x, y)f⁰(x, y) dx dy <+∞,

then, for allT >0, there exists a constantK_T (depending onT,C₀andK(f⁰)) such that sup

t∈[0,T] R³×Y

E(x, y)f (t, x, y) dx dyK_T. (2.33)

Remark 2.1.Forγ >5, we have

R³×Y

1

E^γ(x, y)dx dy <+∞. (2.34)

It will be very useful to show that someL^qbounds off (for 5/6< q <1 andq=s >1) also propagate in time.

2.2. L^qbounds

ObtainingL^qbounds propagation is more technical, that is why we split the proof in several lemmas.

Lemma 2.5.Letβ∈(5/6,1). Then, for allT >0, there exists a constantK_T (depending onT,C₀andK(f⁰))such that

sup

t∈[0,T] R³×Y

f^β(t, x, y) dx dyKT. (2.35)

Proof. Writing

R³×Y

f^β(t, x, y) dx dy=

R³×Y

f^β(t, x, y)E^β(x, y) E^β(x, y)dx dy, we use Young inequality

∀α >1,∀u0, ∀v0, uvu^α α +v^α

α (2.36)

withu=f^β(t, x, y)E^β(x, y),v=_Eβ(x,y)¹ , andα=_β¹>1, and obtain

R³×Y

f^β(t, x, y) dx dyβ

R³×Y

E(x, y)f (t, x, y) dx dy+(1−β)

R³×Y

1 E

β 1−β(x, y)

dx dy.

We conclude with (2.33) and (2.34). 2